48 1. Random Walk and Discrete Heat Equation
Prove the following estimates. The constants c1,c2,c3 are positive
constants independent of N and the estimates should hold for all N
and all j = 1,...,N 1.

β(1,N) β(2,N) · · · β(N 1,N) N
cosh−1(2).
There is a c1 such that
cosh

N
+ cos

N
2
c1 j4
N 4
.
There is a c2 such that
|β(j, N) πj|
c2
j4
N 3
.
There is a c3 such that
β(j, N) c3 j.
Previous Page Next Page